Duplicial functors, descent categories and generalized Hopf modules

Böhm and Ştefan have expressed cyclic homology as an invariant that assigns homology groups ${\mathrm{H}\mathrm{C}}_i^{\chi }(\mathrm{N},\mathrm{M})$ to right and left coalgebras $\mathrm{N}$ respectively $\mathrm{M}$ over a distributive law $\chi$ between two comonads. For the key example associated to a bialgebra $H$, right $\chi$-coalgebras have a description in terms of modules and comodules over $H$. The present article formulates conditions under which such a description is simultaneously possible for the left $\chi$-coalgebras. In the above example, this is the case when the bialgebra $H$ is a Hopf algebra with bijective antipode. We also discuss how the generalized Hopf module theorem by Mesablishvili and Wisbauer features both in theory and examples.